Foundations — van der Waals equation (P + a - V²)(V − b) = RT — physical meaning of a, b

This page assumes you have seen nothing. We will build every letter in the parent equation $\left(P+\frac{a{n}^2}{V^2}\right)(V-nb)=nRT$ from the ground up, one symbol at a time, each one earned before the next appears.

Symbol 0: What is a gas, really?

Before any letter, hold this picture in your head.

Everything in this chapter is about answering: how hard does the swarm push on the walls, and how much room does it need?

Symbol 1: $V$ — volume

Picture: the whole rectangular box in the figure above. If you double the box size, you double $V$.

Why the topic needs it: the ideal law pretends the balls can use all of $V$. van der Waals will say "no — the fat balls block some of it," so we need $V$ first to then subtract from it.

Symbol 2: $P$ — pressure

Picture: every time a ball hits a wall it gives a tiny shove. Millions of shoves per second, added up and averaged over the wall, is pressure.

Why this tool and not "force"? We don't care about one ball's shove — we care about the combined shove per unit area, because that's what a pressure gauge actually reads. Pressure is the right averaged quantity.

Why the topic needs it: attractions make each ball hit softer, lowering $P$. van der Waals will add back the missing pressure. So $P$ is the thing that gets patched.

Symbol 3: $n$ — number of moles (amount of gas)

Picture: $n$ is how many balls are in the box — but counted in convenient bundles (moles) instead of one-by-one, because $6\times 10^{23}$ is too many to count singly.

Why the topic needs it: more gas means more balls taking up space (so more volume blocked, $nb$) and more balls attracting each other (so the $n^2$ in the pressure term). $n$ multiplies both corrections.

Symbol 4: $N_A$ — Avogadro's number

Picture: the bridge between the invisible single-molecule world (radius $r$, tiny volume) and the lab world (litres, moles). Multiply a one-molecule quantity by $N_A$ to get a per-mole quantity.

Why the topic needs it: the excluded volume $b$ is derived per molecule, then scaled up to per mole by multiplying by $N_A$.

Symbol 5: $T$ — temperature

Picture: turn up $T$ and the swarm in the figure speeds up; the balls hit the walls harder and more often.

Why kelvin and not celsius? The gas laws need a scale where $T=0$ means truly no motion. Celsius zero is just "water freezes" — arbitrary. Kelvin zero is the real floor, so ratios like $PV/T$ behave.

Why the topic needs it: at low $T$ the balls fly slowly, so the weak attractions have time to matter — that's exactly when van der Waals' corrections become important.

Symbol 6: $R$ — the gas constant

Picture: think of $R$ as an exchange rate — it converts the "energy of motion" side ($T$, $n$) into the "push × room" side ($P$, $V$) so both sides come out equal.

Why the topic needs it: it's the constant on the right-hand side of the whole equation; without it the units don't match.

Symbol 7: The ideal gas law $PV=nRT$

Now that $P,V,n,R,T$ all exist, we can write the law that van der Waals repairs.

Picture: point-balls with no size and no stickiness, bouncing perfectly elastically. See Ideal Gas Law and Kinetic Theory of Gases for where this comes from.

Why the topic needs it: it is the starting point. van der Waals doesn't throw it away — he edits two symbols inside it.

Symbol 8: $r$ and molecular size — the seed of $b$

Picture: look at the two balls above. The dashed circle of radius $2r$ around one ball is the forbidden zone — no other centre may enter it. That forbidden sphere has volume $\frac{4}{3}\pi (2r{)}^3=8\times$ one ball's volume.

Why the topic needs it: this forbidden zone is the excluded volume. Split $8v$ between the two balls → $4v$ per ball → per mole $b=4N_A\cdot \frac{4}{3}\pi r^3$. This is where the letter $b$ is born.

Symbol 9: Attraction — the seed of $a$

Picture: the inside ball (left) has balanced tugs — arrows cancel. The wall-ball (right) has a net inward tug, so it hits the wall softer — that's lost pressure.

Symbol 10: density $\frac{n}{V}$ and why the $a$-term is squared

Why squared? The pressure loss is a pair effect:

• the number of wall-balls being tugged ∝ density $\frac{n}{V}$, and
• the strength of the tug on each ∝ density $\frac{n}{V}$.

Multiply two density-proportional things → ${\left(\frac{n}{V}\right)}^2$. That's why the correction is $\frac{a{n}^2}{V^2}$, not $\frac{an}{V}$.

Symbol 11: $V_m$ — molar volume

Why the topic needs it: setting $n=1$ turns the messy $\left(P+\frac{a{n}^2}{V^2}\right)(V-nb)=nRT$ into the clean per-mole form $\left(P+\frac{a}{V_m^2}\right)(V_m-b)=RT$ — the form in the parent's title.

Symbol 12: $Z$ — compressibility factor

Picture: $Z$ is just "how far off ideal are we?" as a single number. The whole point of $a$ and $b$ is to explain why $Z$ drifts away from 1.

How it all fits together

Equipment checklist

Cover the right side and test yourself.

Connections

• Yeh note Hinglish mein padho →
• Ideal Gas Law — the law van der Waals repairs
• Kinetic Theory of Gases — where the point-ball picture comes from
• Intermolecular Forces — the source of the attraction constant $a$
• Compressibility Factor Z — the scorecard $Z$
• Critical Constants and Liquefaction — where big $a$ leads
• Boyle Temperature — where $a$ and $b$ effects cancel